Amenable hyperbolic groups

Amenable hyperbolic groups

Pierre-Emmanuel Caprace Université catholique de Louvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium pe.caprace@uclouvain.be Yves de Cornulier Laboratoire de Mathématiques
Bâtiment 425, Université Paris-Sud 11
91405 Orsay
France
yves.cornulier@math.u-psud.fr
Nicolas Monod EPFL
SB – MathGeom – EGG, Station 8
CH–1015 Lausanne, Switzerland
nicolas.monod@epfl.ch
 and  Romain Tessera UMPA, ENS Lyon
46 Allée d’Italie
69364 Lyon Cedex
France
rtessera@umpa.ens-lyon.fr
First draft: February 2012; revised: September 2013.
Abstract.

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a cusp-uniform non-uniform lattice, is very restricted.

Key words and phrases:
Gromov hyperbolic group, locally compact group, amenable group, contracting automorphisms, compacting automorphisms
2010 Mathematics Subject Classification:
Primary 20F67; Secondary 05C63, 20E08, 22D05, 43A07, 53C30, 57S30
F.R.S.-FNRS Research Associate, supported in part by FNRS grant F.4520.11 and the European Research Council
Supported by ANR project “QuantiT” JC08_318197.
Supported in part by the Swiss National Science Foundation and the European Research Council
Supported in part by ANR project GGAA

1. Introduction

1.a. From negatively curved Lie groups to amenable hyperbolic groups

John Milnor [Mil76] initiated the study of left-invariant Riemannian metrics on general Lie groups and observed that a connected Lie group admitting a left-invariant negatively curved Riemannian metric is necessarily soluble; he asked about a more precise characterization. This was answered by E. Heintze [Hei74]: a connected Lie group has a negatively curved left-invariant Riemannian metric if and only if it can be written as a semidirect product , where the group is a (nontrivial) nilpotent Lie group, which is contracted by the action of positive elements of , i.e.  for all .

All these groups thus constitute examples of locally compact groups that are both amenable and (non-elementary) Gromov-hyperbolic. The purpose of the present paper is to study this more general class of groups.

It should be emphasized that, although most works devoted to Gromov-hyperbolicity focus on finitely generated discrete groups, Gromov’s original concept was designed to encompass more general metric groups. We shall mostly focus here on compactly generated locally compact groups; this point of view is in fact very natural, as the full isometry group of a Gromov-hyperbolic metric space might very well be non-discrete. The definition reads as follows: A locally compact group is Gromov-hyperbolic (or, for short, hyperbolic) if it admits a compact generating set such that the associated word metric is Gromov-hyperbolic. In particular hyperbolicity is invariant under quasi-isometries. The definition might look unfamiliar to readers used to deal with locally compact spaces, since the Cayley graph associated with a compact generating set is in general far from locally finite; moreover the natural action of the group on its Cayley graph need not be continuous. This matter of fact is however mitigated by the following characterization, proved in Corollary 2.6 below: a locally compact is Gromov-hyperbolic if and only if it admits a continuous proper cocompact isometric action on a Gromov hyperbolic proper geodesic metric space.

Gromov [Gro87, §3.1,§8.2] divides hyperbolic groups into three classes.

  • The visual boundary is empty. Then is compact.

  • The visual boundary consists of two points. This holds if and only if has an infinite cyclic closed cocompact subgroup. Actually, this can be improved as follows (Proposition 5.6): has a unique maximal compact normal subgroup such that is isomorphic to a cocompact group of isometries of the real line, namely isomorphic to , , , or .

  • The visual boundary is uncountable.

Hyperbolic groups belonging to the first two classes are called elementary and the above description provides for them a largely satisfactory classification; we shall focus on non-elementary hyperbolic groups. For example, a semisimple real Lie group is non-elementary hyperbolic if and only if it has real rank one. All Heintze groups mentioned above are non-elementary hyperbolic.

In order to state our first result, we introduce the following terminology. An automorphism of a locally compact group is called compacting if there is some compact subset such that for each , we have for all sufficiently large . In the special case where for all , we say that is contracting.

The following result provides a first characterization of amenable hyperbolic groups, in the spirit of Heintze’s characterization.

Theorem A.

A locally compact group is amenable and non-elementary hyperbolic if and only if it can be written as a semidirect product or , where is a compacting automorphism of the noncompact group .

We shall give a much more detailled statement in Section 7.B. For now, observe that besides Heintze groups, examples of amenable and non-elementary hyperbolic locally compact groups are provided by the stabilizer of an end in the full automorphism group of a semi-regular locally finite tree. One can also combine a Heintze group with a tree group by some kind of warped product construction, which also yields an example of an amenable non-elementary hyperbolic locally compact group. As a result of our analysis, it turns out that all amenable hyperbolic groups are obtained in this way. For this more comprehensive description of amenable hyperbolic groups, we refer to Theorem 7.3 below. At this point, let us simply mention the following consequence of that description.

Theorem B.

Every amenable hyperbolic locally compact group acts continuously, properly and cocompactly by isometries on a proper, geodesically complete space.

Those spaces will be constructed as fibered products of homogeneous negatively curved manifolds with trees. We call them millefeuille spaces; see §7.A below for a more precise description. Those millefeuille spaces provide rather nice model spaces for amenable hyperbolic groups; one should keep in mind that for general hyperbolic locally compact groups (even discrete ones), it is an outstanding problem to determine if they can act properly cocompactly on any CAT() (or even CAT(0)) space [Gro93, §7.B].

Another consequence of our study is a complete answer to a question appearing at the very end of the paper [KW02] by Kaimanovich and Woess: they asked whether there exists a one-ended locally finite hyperbolic graph with a vertex-transitive group of automorphisms fixing a point at infinity. For planar graphs, this was recently settled in the negative by Georgakopoulos and Hamann [GH12]. We actually show that the answer is negative in full generality.

Corollary C.

If a locally finite hyperbolic graph admits a vertex-transitive group of automorphisms fixing a point at infinity, then it is quasi-isometric to a regular tree and in particular cannot be one-ended.

The proof of Theorem A can be outlined as follows. If a non-elementary hyperbolic locally compact group is amenable, it fixes a point in its visual boundary since otherwise, the ping-pong lemma provides a discrete free subgroup. Notice moreover that since acts cocompactly on itself, it must contain some hyperbolic isometry. The -action on itself therefore provides a special instance of what we call a focal action: namely the action of a group on a hyperbolic space is called focal if fixes a boundary point and contains some hyperbolic isometry. If fixes a point in but does not contain any hyperbolic isometry, then the action is called horocyclic. Any group admits a horocyclic action on some hyperbolic space, so that not much can be said about the latter type. On the other hand, it is perhaps surprising that focal actions are on the contrary much more restricted: for example any focal action is quasi-convex (see Proposition 3.2). In addition, we shall see how to associate canonically a nontrivial Busemann quasicharacter whenever has a focal action on fixing . Roughly speaking, if is a Busemann function at , it satisfies, up to a bounded error, for all . We refer the reader to §3.C for a rigorous definition.

Coming back to the setting of Theorem A, the amenability of implies that the Busemann quasicharacter is actually a genuine continuous character. The fact that an element which is not annihilated by the character acts as a compacting automorphism on the kernel of that character is finally deduced from an analysis of the dynamics of the boundary action, which concludes the proof of one implication.

For the converse implication, we give a direct proof that the Cayley graph of a semi-direct product of the requested form is Gromov-hyperbolic. This part of the argument happens to use only metric geometry, without any local compactness assumption. This approach therefore yields a rather general hyperbolicity criterion, which is stated in Theorem 4.1 below.

1.b. Hyperbolic groups with a cocompact amenable subgroup

We emphasize that, while hyperbolicity is stable under compact extensions, and even under any quasi-isometry, this is not the case for amenability, although amenability is of course invariant under quasi-isometries in the class of discrete groups. Indeed, a noncompact simple Lie group is non-amenable but contains a cocompact amenable subgroup, namely the minimal parabolic . The issue is that is unimodular while is not, so that does not carry any -invariant measure. In particular, the class of hyperbolic locally compact groups containing a cocompact amenable subgroup is strictly larger than the class of amenable hyperbolic locally compact groups. The following result shows that there are however not so many non-amenable examples in that class.

Theorem D.

Let be a non-amenable hyperbolic locally compact group. If contains a cocompact amenable closed subgroup, then has a unique maximal compact normal subgroup , and exactly one of the following holds:

  1. is the group of isometries or orientation-preserving isometries of a rank one symmetric space of noncompact type.

  2. has a continuous, proper, faithful action by automorphisms on a locally finite non-elementary tree , without inversions and with exactly two orbits of vertices, such that the induced -action on the set of ends is -transitive. In particular, decomposes as a nontrivial amalgam of two profinite groups over a common open subgroup.

A locally compact group is called a standard rank one group if it has no nontrivial compact normal subgroup and satisfies one of the two conditions (1) or (2) in Theorem D. Standard rank one groups of type (2) include simple algebraic groups of rank one over non-Archimedean local fields and complete Kac–Moody groups of rank two over finite fields. More exotic examples, and a thorough study in connection with finite primitive groups, are due to Burger and Mozes [BM00].

Any standard rank one group contains a cocompact amenable subgroup, namely the stabilizer of a boundary point, so that the converse of Theorem D holds as well. In fact, several other characterizations of standard rank one groups are provided by Theorem 8.1 below; we shall notably see that they coincide with those noncompact hyperbolic locally compact groups acting transitively on their boundary.

A consequence of Theorem D is that a non-amenable hyperbolic locally compact group which contains a cocompact amenable subgroup is necessarily unimodular. This is a noteworthy fact, since a non-amenable hyperbolic locally compact groups has no reason to be unimodular in general. For example, consider the HNN extension of by the isomorphism between its subgroups and given by multiplication by ; this group is hyperbolic since it lies as a cocompact subgroup in the automorphism group of a -regular tree, but it is neither amenable nor unimodular.

1.c. When are non-uniform lattices relatively hyperbolic?

In a similar way as the concept of Gromov hyperbolic groups was designed to axiomatize fundamental groups of compact manifolds of negative sectional curvature, relative hyperbolicity was introduced, also by Gromov [Gro87], to axiomatize fundamental groups of finite volume manifolds of pinched negative curvature. Several equivalent definitions exist in the literature. Let us only recall one of them, which is the most appropriate for our considerations; we refer the reader to rich literature on relative hyperbolicity for other definitions and comparisons between those (the most relevant one for the definition we chose is [Yam04]). Let be a locally compact group acting continuously and properly by isometries on a hyperbolic metric space . Following P. Tukia [Tuk98, p. 74], we say that the -action (or itself if there is no ambiguity on the action) is cusp-uniform if every boundary point is either a conical limit point or a bounded parabolic point (this notion was introduced by B. Bowditch [Bow99], who called it ‘geometrically finite’). The group is called relatively hyperbolic if it admits some cusp-uniform action on a proper hyperbolic geodesic metric space.

For example, fundamental groups of finite volume manifolds of pinched negative curvature and, in particular, non-uniform lattices in rank one simple Lie groups, are all relatively hyperbolic: their action on the universal cover of the manifold (resp. on the associated symmetric space) is cusp-uniform. Since rank one simple Lie groups are special instances of hyperbolic locally compact groups, one might expect that non-uniform lattices in more general hyperbolic locally compact groups are always relatively hyperbolic. The following result shows that this is far from true.

Theorem E.

Let be a proper hyperbolic geodesic metric space and be a closed subgroup acting cocompactly.

If the action of some non-cocompact closed subgroup on is cusp-uniform, then has a maximal compact normal subgroup such that is a standard rank one group.

Some hyperbolic right-angled buildings, as well as most hyperbolic Kac–Moody buildings, are known to admit non-uniform lattices. Theorem E implies that, provided the building has dimension , these lattices are not cusp-uniform (i.e., they are not relatively hyperbolic with respect to the family of stabilizers of parabolic points).

As remarked above, a non-uniform lattice in a rank one simple Lie group is always relatively hyperbolic. In the case of tree automorphism groups, this is not the case. Necessary and sufficient conditions for a non-uniform lattice have been described by F. Paulin [Pau04]: the key being that a connected fundamental domain for the action of the lattice on the tree has finitely many cusps.

1.d. Amenable relatively hyperbolic groups

As we have seen, there are nontrivial examples of locally compact groups that are both amenable and hyperbolic. We may wonder whether even more general examples might be obtained by considering the class of relatively hyperbolic groups. The following shows that this is in fact not the case.

Theorem F.

Let be an amenable locally compact group. If is relatively hyperbolic, then is hyperbolic.

Organization of the paper

We start with a preliminary section presenting a general construction associating a proper geodesic metric space to an arbitrary compactly generated locally group , together with a continuous, proper, cocompact -action by isometries. This provides a useful substitute for Cayley graphs, which is better behaved since it avoids the lack of continuity and local compactness that Cayley graphs may have in the non-discrete case. Since is quasi-isometric to , the hyperbolicity of the former is equivalent to the hyperbolic of the latter.

The proofs of the main results are then spread over the rest of the paper, and roughly go into three steps. The first part consists in a general study of isometric actions on hyperbolic spaces, culminating in the proof of Theorem 4.1 which implies that certain groups given as semi-direct products with cyclic factor are hyperbolic. This part is mostly developed in a purely metric set-up, without the assumption of local compactness. It occupies Section 3 and Section 4, and yields the implication from right to left in Theorem A.

In Section 5, we start making local compactness assumptions, but yet not exploiting any deep structural result about locally compact groups. This is where Theorem F is proven. This chapter moreover provides the implication from left to right in Theorem A, whose proof is thus completed in Section 6.A.

Finally, a more comprehensive version of Theorem A, as well as its corollaries, is proved in §7.B after some preliminary work about the structure of groups admitting compacting automorphisms in Section 6, and on the construction of millefeuille spaces in §7.A. Similarly, a more comprehensive version of Theorem D is stated and proved in §8.A. Theorem E is then easily deduced in the next subsection.

Acknowledgement

We thank the referee, whose comments were helpful in improving the presentation of the paper.

2. Preliminaries on geodesic spaces for locally compact groups

It is well-known that a topological group with a proper, cocompact action by isometries on a locally compact geodesic metric space is necessarily locally compact and compactly generated. It turns out that the converse is true, and that the space can be chosen to be a piecewise-manifold. This is the content of Proposition 2.1 below. Its relevance to the rest of the paper is through Corollary 2.6. The remainder of the section is devoted to its proof and is independent from the rest of the paper, so the reader can, in a first reading, take the proposition and its corollary for granted and go directly to Section 3.

Proposition 2.1.

Let be a compactly generated, locally compact group. There exists a finite-dimensional (in the sense of topological dimension) locally compact geodesic metric space with a continuous, proper, cocompact -action by isometries.

In fact is a connected locally finite gluing of Riemannian manifolds along their boundaries.

An immediate consequence is the fact that a closed cocompact subgroup of a compactly generated locally compact group is itself compactly generated. This is well-known and can be established more simply by a direct algebraic argument, see [MŚ59].

Let us begin by illustrating Proposition 2.1 with significant examples.

  • When is discrete, we consider its Cayley graph with respect to a finite generating set.

  • When is a connected Lie group, is taken as endowed with a left-invariant Riemannian metric.

  • When is an arbitrary Lie group, we pick a finite subset whose image in is a generating subset and endow the (non-connected) manifold with a left-invariant Riemannian metric; then for each coset of and each , we consider a strip (with the product Riemannian metric) and attach it to by identifying to and to . The resulting space is path-connected and endowed with the inner length metric associated to the Riemannian metric on each strip.

  • When is totally disconnected, a Cayley–Abels graph construction is available, generalizing the discrete case. It goes back to Abels [Abe74, Beispiel 5.2]. It consists in picking a compact generating subset that is bi-invariant under the action of some compact open subgroup , considering the (oriented, but unlabeled) Cayley graph of with respect to , and modding out by the right action of . The resulting graph is locally finite; the action of is continuous, vertex-transitive and proper, the stabilizer of the base-vertex is .

The general case is a common denominator between the latter two constructions. Roughly speaking, we construct by fibering a Lie quotient associated to over a Cayley–Abels graph for . The following classical theorem allows to bypass some of the technical difficulties.

Theorem 2.2 (H. Yamabe).

Let be a connected-by-compact locally compact group. Then is compact-by-(virtually connected Lie).

(By convention (A)-by-(B) means with a normal subgroup satisfying (A) so that the quotient group satisfies (B).)

Proof.

See Theorem 4.6 in [MZ55]. ∎

Lemma 2.3.

Let be any locally compact group. There is a compact subgroup whose image in is open. If is Lie, then we can assume that .

Proof of Lemma 2.3.

By van Dantzig’s theorem [vD31, p. 18], the totally disconnected group contains a compact open subgroup; let be the pre-image in of that subgroup, so that is open in and is compact.

Thus Yamabe’s theorem applies and contains a compact normal subgroup such that is a Lie group. Since is compact, it follows that has finitely many connected components and therefore, upon replacing by a smaller open subgroup, we can assume that is a connected Lie group. The next lemma, of independent interest, implies in particular that we have ; thus indeed the image of in is the open subgroup .

For the additional statement, assume that is Lie. Then there exists a neighbourhood of 1 such that contains no nontrivial subgroup. Since any compact group is pro-Lie (by Peter-Weyl’s Theorem), contains a normal subgroup of such that is Lie; clearly we have . If is the projection to , it follows that is both Lie and profinite, hence is finite, so is open in as well. ∎

Lemma 2.4.

Let be a locally compact group with a quotient map onto a Lie group . Then .

Proof.

Obviously (considering ) we can suppose that is connected. We have to show that , where is the kernel of . Since is both a quotient of the connected group and of the totally disconnected group , it is trivial, in particular is dense. This allows to conclude at least when is compact or is a Lie group (not assumed connected). Indeed, in both cases this assumption implies that is closed (when is a Lie group, see this by modding out by ).

In general, let be an open subgroup of such that is compact. Since is an open map, is an open subgroup of and therefore is equal to . This allows to assume that is connected-by-compact. By Yamabe’s theorem, thus has a maximal compact normal subgroup . We can factor as the composition of two quotient maps . The left-hand map has compact kernel, and has a continuous injective map into and therefore is a Lie group. So the result follows from the two special cases above. ∎

Remark 2.5.

The assumption that is a Lie group is essential in Lemma 2.4. Indeed, let , where denotes the (compact) additive group of the -adic integers. Let be a copy of embedded diagonally in , and let be the quotient group. The group is the so-called solenoid and can alternatively be defined as the inverse limit of the iterated -fold covers of the circle group. It is connected (but not locally arcwise connected). The image of under the quotient map is dense, but properly contained, in .

Proof of Proposition 2.1.

Let be a compactly generated locally compact group. Upon modding out by the unique maximal compact normal subgroup of , we can assume that is a Lie group and endow it with a left-invariant Riemannian metric. Set . By Lemma 2.3, there is a compact subgroup whose image in is open and .

Let be a compact generating set with . Since is open, is finite and we pick a finite set of representatives . We define for

We recall that a Cayley–Abels graph for is given by the discrete vertex set and the (oriented edge)-sets , with natural -action and source and target maps. On the other hand, there are canonical surjective -maps . The fibres of these maps are , which are indeed connected Riemannian manifolds. By construction, the -action is compatible with the gluings of boundary components of determined by the map to the Cayley–Abels graph, and we endow the resulting connected space with the inner length metric associated to the Riemannian structure. Explicitly, the gluing is generated by identifying with whenever and with whenever . ∎

The above construction is of course much more general than what is needed in the present article. Indeed, as a consequence of the results of the article, a hyperbolic locally compact group has a continuous proper cocompact action either on a millefeuille space (this includes the special case when this space is simply a homogeneous negatively curved manifold), or on a connected graph. Actually, the latter description shows that the space can be chosen in addition to be contractible: indeed, in the case of a connected graph, the Rips complex construction as described in [Gro87, 1.7.A] is applicable.

For the time being, we only record the following consequence of Proposition 2.1.

Corollary 2.6.

For a locally compact group , the following are equivalent.

  1. is hyperbolic, i.e. compactly generated and word hyperbolic with respect to some compact generating set.

  2. has a continuous proper cocompact isometric action on a proper geodesic hyperbolic space.∎

3. Actions on hyperbolic spaces

«  Ça faut avouer, dit Trouscaillon qui, dans cette simple ellipse, utilisait hyperboliquement le cercle vicieux de la parabole.  »

(R. Queneau, Zazie dans le métro, 1959)

After reviewing some basic features of groups acting on hyperbolic spaces, the goal of this section is to highlight the importance of focal actions (see Section 3.A below for the precise definitions). Indeed, while actions of general type have been studied in thorough detail in a myriad of papers on hyperbolic spaces, other actions, sometimes termed as “elementary”, have been considered as uninteresting. Notably, and as a consequence of an inadequate terminology, the distinction between horocyclic and focal actions has been eclipsed. Several basic results in this section (especially Proposition 3.2, Lemma 3.4, and Proposition 5.5) illustrate how different these two types of actions are and how essential it is to take a specific look at focal actions.

Throughout this section, we let be a Gromov-hyperbolic geodesic metric space.

Recall that is called proper if closed balls are compact; due to the Hopf–Rinow theorem for length spaces, it is equivalent to require that be locally compact and complete. Recall further that the full isometry group , endowed with the compact open topology, is a second countable locally compact group. We emphasize that will not be assumed proper, unless explicitly stated otherwise.

3.a. Gromov’s classification

The material in this section follows from [Gro87, 3.1]. Let be an abstract group, and consider an arbitrary isometric action of on a nonempty hyperbolic geodesic metric space .

The visual boundary (or boundary) of is defined as follows. Fix a basepoint in , define the norm and the Gromov product

Note that A sequence in is Cauchy-Gromov if tends to infinity when both tend to ; by the previous inequality, this does not depend on the choice of . We identify two Cauchy-Gromov sequences and if tends to infinity. This is indeed an equivalence relation if is -hyperbolic, in view of the inequality

whose validity is a definition of -hyperbolicity (if it holds for all ). The boundary is the quotient set of Cauchy-Gromov sequences by this equivalence relation. (In other words, consider the uniform structure given by the entourages as ranges over ; then is the completion from which the canonical image of has been removed.)

If is a group acting on by isometries, the boundary , also called the limit set of in , consists of those elements in the boundary, that can be represented by a Cauchy-Gromov sequence of the form with . Since and are equivalent for all , this does not depend on . The action of induces an action on , which preserves the subset .

A crucial case is when is generated by one isometry . Recall that is called

  • if it has bounded orbits;

  • if it has unbounded orbits and ;

  • if .

The above limit always exists by subadditivity, and the definition clearly does not depend on the choice of . Also, it is straightforward that if preserves a geodesic subset , then the type of is the same as the type of . In terms of boundary, it can be checked [CDP90, Chap. 9] that

  • is elliptic is empty;

  • is parabolic is a singleton;

  • is hyperbolic consists of exactly two points.

For an action of an arbitrary group , Gromov’s classification [Gro87, 3.1] goes as follows. The action is called

  • elementary and

    • if orbits are bounded;

    • if it is unbounded and has no hyperbolic element;

    • if it has a hyperbolic element and any two hyperbolic elements have the same endpoints;

  • non-elementary111We follow Gromov’s convention. It turns out that in the special case of proper actions of discrete groups, focal actions do not exist and thus elementary actions are precisely those with a finite orbit on the boundary. For this reason, Gromov’s conventions were misinterpreted by several authors, who unaccurately consider the focal case as elementary. and

    • if it has a hyperbolic element, is not lineal and any two hyperbolic elements have a common endpoint (it easily follows that there is a common endpoint for all hyperbolic elements);

    • if it has two hyperbolic elements with no common endpoint.

These conditions can be described in terms of the boundary .

Proposition 3.1.

The action of is

  • bounded if and only if is empty;

  • horocyclic if and only if is reduced to one point; then is the unique finite orbit of in ;

  • lineal if and only if consists of two points; then contains all finite orbits of in ;

  • focal if and only if is uncountable and has a fixed point in ; then is the unique finite orbit of in ;

  • of general type if and only if is uncountable and has no finite orbit in .

In particular, the action is elementary if and only if has at most two elements, and otherwise is uncountable.

Sketch of proof.

If the action is horocyclic, the proof of [CDP90, Theorem 9.2.1] shows that for every sequence such that tends to infinity, the sequence is Cauchy-Gromov; it follows that is a singleton. It follows in particular that the intersection of an orbit with any quasi-geodesic is bounded.

If and has another finite orbit on the boundary, then we can suppose that it has another fixed point by passing to a subgroup of finite index. Let us consider a (metric) ultrapower of ; namely is obtained as follows: endow the space of bounded sequences in with the pseudo-distance defined as the limit of the distances along a non-principal ultrafilter; then is the metric space obtained by identifying sequences at pseudo-distance zero. It admits a canonical isometric embedding of ; it is also a geodesic metric space and is hyperbolic with the same hyperbolicity constant, and the -action canonically extends to an action . There is an natural inclusion ; since is -invariant, it follows that and in particular, the type of the action on is the same as the type of the action on , i.e., horocyclic. Moreover, any pair of distinct points in can be joined by a geodesic in . Consider a geodesic in joining and . Its -orbit is a -invariant quasi-geodesic. Since the action is horocyclic, the above remark shows that the action of on this quasi-geodesic, and hence on , is bounded, a contradiction.

The other verifications are left to the reader (the uncountability of in the non-elementary cases follows from Lemma 3.3). ∎

3.b. Basic properties of actions and quasi-convexity

As before, is a hyperbolic geodesic space, without properness assumptions. Recall that a subset is quasi-convex if there exists such that for all there exist a sequence in with for all and . We say that an action is quasi-convex if some (and hence every) orbit is quasi-convex. If the acting group is locally compact and the action is metrically proper, this is equivalent to the requirement that is compactly generated and undistorted in (i.e. the orbit map is a quasi-isometric embedding for some/all ).

The notions of horocyclic and focal actions, i.e. those unbounded actions with a unique fixed point at infinity, are gathered under the term of quasi-parabolic actions in [Gro87, KN04], while horocyclic actions were termed parabolic. Nevertheless, the following proposition, which does not seem to appear in the literature, shows that horocyclic and focal actions exhibit a dramatically opposite behaviour.

Proposition 3.2.

If the action of is bounded, lineal or focal, then it is quasi-convex. On the other hand, a horocyclic action is never quasi-convex, while an action of general type can be either quasi-convex or not.

Proof.

The bounded case is trivial and in the lineal case, preserves a subset at bounded Hausdorff distance of a geodesic and is thus quasi-convex.

Assume that the action is focal with as a global fixed point. We have to prove that some given orbit is quasi-convex. Let be a hyperbolic element, and let be a point on a geodesic line between the two fixed points and of (embed if necessary into a (metric) ultrapower as in the proof of Proposition 3.1 to ensure the existence of this geodesic). The -orbit of is a discrete quasi-geodesic. Observe that the orbit is the union of quasi-geodesics , with varying in . In particular, contains quasi-geodesics between all its points to . Now, let and be two points in . Recall that given a quasi-geodesic triangle between three points in the reunion of a hyperbolic space with its boundary, the union of two edges of this triangle is quasi-convex. Applying this to , and , we see that a quasi-geodesic between and can be found in the orbit , which is therefore quasi-convex.

If the action of is horocyclic, we observed in the proof of Proposition 3.1 that the intersection of any orbit with a quasi-geodesic is bounded. More precisely, for every there exists (depending only on and ) such that the intersection of any orbit and any -quasi-geodesic is contained in the union of two -balls. If the action is quasi-convex, given there exists such that any two points in are joined by a -quasi-geodesic within ; taking two points at distance we obtain a contradiction.

For the last statement, it suffices to exhibit classical examples: for instance has a proper cocompact action on a tree, but its action on the hyperbolic plane is not quasi-convex. ∎

Let act on by isometries. Recall that a Schottky subsemigroup, resp. subgroup, for the action of on is a pair such that the orbit map , is a quasi-isometric embedding of the free semigroup (resp. subgroup) on . An elementary application of the ping-pong lemma [Gro87, 8.2.E, 8.2.F] yields the following.

Lemma 3.3.

If the action of is focal (resp. general type), then there is a Schottky subsemigroup (resp. subgroup) for the action of on .∎

It is useful to use a (metrizable) topology on . A basis of neighbourhoods of the boundary point represented by the Cauchy-Gromov sequence is

Gromov shows [Gro87, 8.2.H] that if the action is of general type, then the actions of on and on are topologically transitive. This very important (and classical) fact will not be used in the paper. On the other hand, in the focal case, we have the following observation, which, as far as we know, is original.

Lemma 3.4.

Let the isometric action of on be focal with as a fixed point. Then the action of on is topologically transitive.

Proof.

By Proposition 3.2, there is no loss of generality in assuming that the action of on is cobounded, so that is covered by -balls around points of an orbit for some . Fix a point and an open subset in . For some there exists a -quasi-geodesic in joining and ; let be one of its points. There exists a -ball so that every -quasi-geodesic with endpoint and passing through has its second endpoint in . There exists such that belongs to . It follows that the second endpoint of , which equals , lies in . ∎

Lemma 3.5.

Let the isometric action of on be horocyclic with fixed point in . Fix and endow with the left-invariant (pseudo)distance .

Then the action of on satisfies the following property (akin to metric properness): for every closed subset of not containing , the set is bounded in .

Proof.

We can replace by a (metric) ultrapower (see the proof of Proposition 3.1), allowing the existence of geodesics between any two points (at infinity or not).

Let be a closed subset of not containing . Note that there exists a ball of radius such that every geodesic whose endpoints are and some point in , passes through . Fix such that . In particular, there exists a geodesic such that both and are issued from and pass through . It follows that before hitting , and lie at bounded distance (say, ) from each other. Fix some and let be the geodesic ray in joining to . Then either is contained in the -neighbourhood of , or vice versa. Since is not hyperbolic, using the inequality (see [CDP90, Lemma 9.2.2]) it easily follows (using that , and are close to the geodesic ray and the equality ) that . Thus the set is -bounded, hence -bounded. ∎

Define the bounded radical of as the union of all normal subgroups that are -bounded, i.e. such that the action of on is bounded.

Lemma 3.6.

Suppose that the -action is not horocyclic. Then the following properties hold.

  1. The action of on is bounded.

  2. If moreover the -action is neither lineal, is equal to the kernel of the action of on .

Proof.

If is an -bounded normal subgroup, then is bounded, and therefore the are uniformly bounded when ranges over an orbit . In particular, the action of on is trivial. This proves the inclusion (without assumption on the action).

Note that the action of on is trivial, so by Proposition 3.1, the action of is bounded, lineal or horocyclic; Lemma 3.5 then shows that the action of cannot be horocyclic.

  • If the action of on is bounded, it follows from the definition of that , and both (a) and (b) follow.

  • If the action of on is lineal, its 2-element boundary is preserved by and therefore the action of is lineal as well (so we do have to consider (b). Since consists of elliptic isometries, its action is either bounded or horocyclic, but since a horocyclic action cannot preserve a 2-element subset in by Proposition 3.1, the action of cannot be horocyclic and therefore is bounded, so (a) is proved.

3.c. Horofunctions and the Busemann quasicharacter

The material in this section is based on [Gro87, 7.5.D] and [Man08, Sec. 4].

Let be an arbitrary hyperbolic space and be a point at infinity. We define a horokernel based at to be any accumulation point (in the topology of pointwise convergence) of a sequence of functions

where is any sequence in converging to , i.e. a Cauchy-Gromov sequence representing . By the Tychonoff theorem, the collection of all horokernels based at is non-empty; it consists of continuous functions, indeed -Lipschitz in each variable. Moreover, any horokernel is antisymmetric by definition. For definiteness, many authors propose the gordian definition of the Busemann kernel of as the supremum of (losing continuity and antisymmetry in general); it turns out that it remains at bounded distance of any horokernel, the bound depending only on the hyperbolicity constant of , see § 8 in [GH90]. Let us also mention the notion of horofunction , which depends on the choice of a basepoint .

Recall that a function defined on a group is a quasicharacter (also known as quasimorphism) if the defect

is finite; it is called homogeneous if moreover for all and ; in that case, is constant on conjugacy classes. Given an isometric group action on fixing , there is a canonical homogeneous quasicharacter associated to the action, which was constructed by J. Manning [Man08, Sec. 4]. The following is a variant of an idea appearing in T. Bühler’s (unpublished) Master’s thesis.

Proposition 3.7.

Let be a locally compact group acting continuously by isometries on . Let , and . Then the function

is a well-defined continuous homogeneous quasicharacter, called Busemann quasicharacter of , and is independent of and of .

Moreover, the differences and are bounded (the bound depending only on the hyperbolicity constant of ).

Proof.

By a direct computation, we have

since is fixed by , the latter quantity is bounded by a constant depending only on the hyperbolicity of . Therefore, the function is a continuous quasicharacter. Given any quasicharacter on a group , it is well-known that for all , the sequence converges (because the sequence is subadditive, where the constant is the defect) and that the limit is a homogeneous quasicharacter (by an elementary verification). This limit is the unique homogeneous quasicharacter at bounded distance from and a bounded perturbation of yields the same limit. Returning to our situation, it only remains to justify that the limit is continuous. It is Borel by definition, and any Borel homogeneous quasicharacter on a locally compact group is continuous [BIW10, 7.4]. ∎

The Busemann quasicharacter is useful for some very basic analysis of boundary dynamics:

Lemma 3.8.

Let act on by isometries and let be a boundary point. Then the (possibly empty) set of hyperbolic isometries in is , and the set of those with attracting fixed point is . In particular, the action of is bounded/horocyclic if and only if , and lineal/focal otherwise.

Proof.

Elements of acting as hyperbolic isometries with attracting fixed point satisfy , as we see by direct comparison with horokernels. In particular all elements acting as hyperbolic isometries satisfy .

Conversely, if satisfies , then being linear in , the sequence is a quasi-geodesic with the -endpoint at . By hyperbolicity of , it follows that is a hyperbolic isometry with attracting fixed point . It also follows that if satisfies then it acts as a hyperbolic isometry. ∎

The Busemann quasicharacter is particularly nice in connection with amenability: indeed, as a corollary of Proposition 3.7, we deduce the following.

Corollary 3.9.

Let be a locally compact group acting continuously by isometries on and fixing the boundary point . Assume that is amenable, or that is proper.

Then the Busemann quasicharacter is a continuous group homomorphism (then called Busemann character) at bounded distance of independently of .

Proof of Corollary 3.9.

The well-known fact that a homogeneous quasicharacter of an amenable group is a homomorphism can be verified explicitly by observing that one has when is an invariant mean on .

If is amenable, this applies directly. Now assume that is proper. If the result is trivial, so assume that the action is lineal/focal. Since is proper, we can suppose that (and thus acts properly) and we argue as follows. By Proposition 3.2, sufficiently large bounded neighbourhoods of a -orbit are quasi-geodesic. Therefore, replacing by such a subset if necessary, we can assume that the -action on is cocompact, so that Lemma 3.10 below applies to show that is amenable. We are thus reduced to the previous case. ∎

A metric space is called quasi-geodesic if there is some constant such that any two points can be joined by a -quasi-geodesic. For example, any orbit of a quasi-convex group of isometries in a hyperbolic metric space is quasi-geodesic. Remark that it is always possible to embed a quasi-geodesic subset coboundedly into a geodesic space (by gluing geodesic paths), but it is delicate to get a proper geodesic space.

Lemma 3.10 (S. Adams).

Let be a proper quasi-geodesic hyperbolic space having a cocompact isometry group (or more generally, having bounded geometry), then for every , the stabilizer is amenable.

Proof.

See [Ada96, 6.8]; another simpler proof can be found in [Kai04]. Remark that in [Ada96], the result is stated assuming is geodesic, but quasi-geodesic is enough with no alteration of the proof, and is a more robust statement because it allows passing to a quasi-convex subset. ∎

Remark 3.11.

If is proper, one can also use the amenability of to prove the existence of a (non-canonical) -invariant function at bounded distance of the horokernels based at . Indeed, the Tychonoff theorem implies that is compact and the desired function is obtained by integrating an invariant measure on . More generally, one can use the amenability of the -action on to make an -equivariant choice of such Busemann-like functions, depending measurably on .

Example 3.12.

There are many instances where the Busemann quasicharacter is a character: the trivial case of bounded/horocyclic actions, and also the case when is proper, by Corollary 3.9. It is also the case when the hyperbolic space is CAT(0): indeed in this case there is a unique horokernel based at each . Therefore, this unique horokernel coincides with the Busemann function and moreover we then have for all and all .

On the other hand, here is an example of an oriented lineal action where the Busemann character is not a homomorphism. Consider the centralizer in of the translation . This can be interpreted as the universal covering of the group of oriented homeomorphisms of the circle, and we thus denote it by . Endow with the structure of Cayley graph with respect to the generating set , i.e. with the incidence relation if . This incidence relation is preserved by the action of , which thus acts on the Cayley graph. This Cayley graph is obviously quasi-isometric to and this action is transitive and lineal. If , then is the translation number , which is a classical example of non-homomorphic homogeneous quasicharacter. Actually, in restriction to some suitable subgroups (e.g. the inverse image of in ), this quasicharacter remains non-homomorphic and this provides examples where the acting group is finitely generated.

Here is now an example of a focal action, based on the same group. Set (or any nontrivial finite group). Consider the permutational wreath product , where acts by shifting the indices in according to its action on . Let be the subgroup of elements with support in and let be the set of elements with translation number in (i.e. those elements such that ). Then is generated by and the corresponding Cayley graph is hyperbolic, the action of being focal. To see this, first observe that if be the translation , then is cobounded in and thus it suffices to check that , with the word metric associated to the generating set , is hyperbolic focal. But this is indeed the case by Theorem 4.1.

4. Focal actions and confining automorphisms

Recall from the introduction that the action of a group on a hyperbolic metric space is called focal if it fixes a unique boundary point and if some element of acts as a hyperbolic isometry. Let the associated Busemann quasicharacter, as in §3.C. The action of is said to be regular focal if is a homomorphism. This holds in particular if is CAT(0), or if is proper (i.e. if balls are compact). The latter case will in fact crucial in the proof of Theorem 7.3. Example 3.12 illustrates that a focal action need not be regular in general.

Let be a group (with no further structure a priori) and let be an automorphism of and a subset of . We say that the action of is [strictly] confining into (we omit when no ambiguity incurs) if it satisfies the following three conditions

  • is [strictly] contained in ;

  • ;

  • for some non-negative integer .

In case is a locally compact group, there is a close relation between confining and compacting automorphisms, which will be clarified in Corollary 6.2 below.

Notice that the group is generated by the set . Endow with the word metric associated to . Given an action of a group on a metric space and a point , define a pseudo-metric on by .

Theorem 4.1.

Let be a group with a cobounded isometric action on a geodesic metric space . Then the following assertions are equivalent.

  1. is hyperbolic and the -action is regular focal;

  2. There exist an element and a subset such that

    • the image of in has infinite order;

    • the action of on is confining into ;

    • setting and , the inclusion map is a quasi-isometry for some (hence every) .

Moreover if (ii) holds, the Busemann character in (i) is proportional, in restriction to , with the obvious projection to .

The implication (ii)(i) includes the fact that for every as above, is Gromov-hyperbolic (see Proposition 4.6); this remains true when but in this case is elementary hyperbolic and quasi-isometric to the real line.

Beyond the locally compact case, a simple example of a group as above is a Banach space, being the unit ball and being the multiplication by some positive scalar .

4.a. From focal actions to focal hyperbolic groups

The following proposition reduces the proof of Theorem 4.1 to a statement in terms of metric groups: A group is regular focal if and only if it has a subgroup with a semidirect decomposition and a subset so that is confining into and the inclusion map is a quasi-isometry.

Proposition 4.2.

Let be a group acting by isometries on a hyperbolic metric space , and let . Let be any left-invariant pseudo-metric on such that the orbit map is a quasi-isometry. Then

  • the action is focal if and only if is hyperbolic and the left -action on is focal;

  • the action is regular focal if and only if is hyperbolic and the left -action on is regular focal.

Let us start with two useful lemmas.

Lemma 4.3.

Let be a homogeneous quasicharacter on a group . Suppose that is bounded in restriction to some normal subgroup . Then induces a (homogeneous) quasicharacter on . In particular if all homogeneous quasicharacter of are characters, then so is .

Proof.

Let and . Using that is bounded on , we get

for some and some bounded . Letting tend to infinity, we obtain that , which proves the lemma. ∎

Lemma 4.4.

Let and be homogeneous quasicharacters on a group such that

for some . If is a character, then so is .

Proof.

Since and are homogeneous, we have . Let . By the previous lemma, both and induce a homogenous quasicharacter of the quotient , and their value on is completely determined by their value on . Since the quotient is abelian, all its homogeneous quasicharacters are characters, and the claim follows. ∎

Proof of Proposition 4.2.

Recall that the orbits of a focal action are quasi-convex (Proposition 3.2), hence are quasi-geodesic subspaces of . Therefore, the fact that the action is focal (resp. regular focal) or not can be read on the restriction of the action on one orbit of . In other words, this proves the proposition when is exactly the distance induced by the orbit map. Now we need to prove that being focal (resp. regular focal) for a metric group only depends on a choice of metric up to quasi-isometry. This is clear for focal, since a quasi-isometry induces a homeomorphism between the boundaries, and therefore does not change the dynamics of the -action on its boundary. The quasi-isometric invariance of the regularity condition follows from Lemma 4.4. ∎

4.b. From regular focal groups to confining automorphisms

The implication (i)(ii) in Theorem 4.1 will be deduced from the following.

Proposition 4.5.

Let be a regular focal hyperbolic metric group. Let be the unique fixed point of the boundary and be the corresponding Buseman character. Set and let . Then

  • is a cobounded, normal subgroup of . (In particular, if is locally compact and the action continuous and proper, it is a cocompact normal subgroup of .)

  • There exist satisfying: for all there exists such that for all , In particular, is confining into .

Proof.

We start with a preliminary observation. Since acts as a hyperbolic isometry, the focal point must be either its attracting or repelling fixed point. Upon replacing by , we may assume that it is the attracting one. In particular, the sequence defines a quasi-geodesic ray tending to . Therefore, so is the sequence for any . Recall that here is a constant (depending only on the hyperbolicity constant of ) such that any two quasi-geodesic ray with the same endpoint are eventually -close to one another. In particular, if , then for all larger than some , where depends only on .

We now turn to the assertion (i). The only nontrivial statement is that is cobounded.

Let thus be arbitrary, and let be such that . By the preliminary observation, we have . Therefore we have

Since , this proves that is -dense in , as desired.

The assertion (ii) also follows from the preliminary observation, since for all , we have . ∎

4.c. From confining automorphisms to hyperbolic groups

We now turn to the converse implication in Theorem 4.1, which is summarized in the following proposition.

Proposition 4.6.

Let be a group and let be an automorphism of which confines into some subset . Let . Then the group is Gromov-hyperbolic with respect to the left-invariant word metric associated to the generating set . If the inclusion is strict, then it is focal.

Upon replacing by , we may assume that is symmetric and contains . The group is endowed with the word metric associated with the symmetric generating set . Remark that this metric is -geodesic, in the sense that for all at distance from one another, there exists a so-called -geodesic between them, i.e.  such that . Denote by the -ball in this metric.

The following easy but crucial observation is a quantitative version of the fact that unbounded horocyclic actions are always distorted, see Proposition 3.2.

Lemma 4.7.

There exists a positive integer such that all 1-geodesics of contained in have length .

Proof.

Actually, we will prove a stronger statement, which, roughly speaking, says that is exponentially distorted inside . Note that

Since , we infer, more generally that

Hence, if there exists a 1-geodesic of length contained in , then must satisfy , which obviously implies that it is bounded by some number depending only on (say, ). ∎

Now let us go further and describe 1-geodesics in . Observe that a 1-geodesic between and can be seen as an element in the free semigroup over of minimal length representing (note that in this semigroup we do not have ; the reason we work in the free semigroup rather than free group is that the loop is not viewed as at bounded distance to the trivial loop).

Lemma 4.8.

Every path emanating from , of the form with and , is at distance  from a path of the form with and .

In particular, every -geodesic from to in is at uniformly bounded distance from a word of the form with , , and for all , where is the constant from Lemma 4.7.

Proof.

Let be a word in representing a path joining to some , such that and . Note that every subword of the form (resp. ), with , and can be replaced by (resp. ), with . Such an operation moves the -geodesic to another -geodesic at distance one (because for all ).

With this process we can move positive powers of all the way to the right, and negative powers to the left, obtaining after at most operations a minimal writing of as , with , and for all . This proves the first assertion.

Assuming now that is of minimal length among words representing . In the above process of moving around powers of , the length of the word never gets longer, and since is minimal, it cannot get shorter either. Since the word has minimal length, it forms a geodesic in and we deduce from Lemma 4.8 that and is at distance at most from . ∎

Proof of Proposition 4.6.

Consider a 1-geodesic triangle in . By Lemma 4.8, we can suppose that is of the form

where are words of length in and . Since forms a loop, its image under the projection map onto is also a loop, hence we have .

Let us prove that is thin, in the sense that every edge of the triangle lies in the -neighborhood of the union of the two other edges. Note that if after removing a backtrack in a triangle (in terms of words this means we simplify ), we obtain a -thin triangle, then it means that the original triangle was -thin. Upon removing the three possible backtracks, permuting cyclically the edges, and changing the orientation, we can suppose that

with and .

By Lemma 4.8, the -path is a distance from a -geodesic segment of the form (where has length ), and removing a backtrack by replacing by we get a triangle